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f(x) = $\{\begin{array}{c}\frac{x}{|x|},ifx<0\\ -1,ifx\ge 0\end{array}$

If ${\text{cot}}^{-1}\left[{\left(\text{cos}\alpha \right)}^{1/2}\right]-{\text{tan}}^{-1}\left[{\left(\text{cos}\alpha \right)}^{1/2}\right]=x,$ then $\sin x$ equals

If from a point $P,$ tangents $PQ$ and $PR$ are drawn to the ellipse $\frac{x^2}{2}+y^2=1,$ such that equation of $QR$ is $x+3y=1,$ then the coordinates of $P$ is

If $\int \ln (x^2+x)dx=x\ln (x^2+x)+f\left(x\right)+c$, then $f\left(x\right)$ equals:

If the function $f\left(x\right)=\{\begin{array}{cc}x+2& \text{if}x<2\\ a{x}^2+bx+3& \text{if}2\le x<3\\ 2x+a+b& \text{if}x\ge 3\end{array}$ is continous, then the value of $(a^2+b^2)$ is

Let $\overrightarrow{a}$ be a unit vector and $\overrightarrow{b}$ be a non-zero vector not parallel to $\overrightarrow{a}$. If two sides of a triangle are represented by the vectors $\sqrt{3}(\overrightarrow{a}\times \overrightarrow{b})$ and $\overrightarrow{b}-(\overrightarrow{a}\cdot \overrightarrow{b})\overrightarrow{a}$, then the angles of the triangle are

$\int \frac{dx}{(x+1)\sqrt{1+x-x^2}}$ equals to (where $C$ is integration constant)

If A = { a, b } and B = { 1, 2 } be two sets , then the number of relations that can be formed from the set A to set B is

If $f\left(x\right)=(x^2-1)(3+x^2{)}^{\frac{1}{2}}(8+x^4{)}^{\frac{1}{2}},$ then $f^{\prime }\left(1\right)$ is equal to

Show that the function f Defined by f(x)=|1-x+|x|| is a continuous function

If $x=\tan (22.5{)}^{\circ },$ then the value of $x^4+3x^3-3x^2-9x+6$ is

The largest integer ' n ' which makes also an integer is

If n is positive integer and three consecutive coefficients in the expansion of $(1+x{)}^n$ are in the ratio 6 : 33 : 110, then n =

The area bounded by the curves $y=x\ln x$ and $y=2x-2x^2$ is

The values of $a$ for which the number $6$ lies in between the roots of the equation $x^2+2(a-3)x+9=0,$ belong to

The equation of the median through the vertex $A$ of triangle $ABC$ whose vertices are $A(2,5),B(-4,9)$ and $C(-2,-1)$ is

Integrate the following functions.
$\int \frac{dx}{\sqrt{1+4x^2}}.$

If the system $\left[\begin{array}{ccc}1& \alpha & {\alpha }^2+4\\ 1& \beta & {\beta }^2+4\\ 1& 4& 20\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ 2\\ -1\end{array}\right]$ of linear equations has unique solution, then which of the following is true

$(\text{where}\alpha ,\beta \in R)$

Write the range of the function
$f\left(x\right)=cos\left[x\right],where\frac{-\pi }{2}<x<\frac{\pi }{2}$

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 49y² – 16x² = 784

The probability distribution of a random variable X is given below:

$\begin{array}{ccccc}X=x& 0& 1& 2& 3\\ P(X-x)& \frac{1}{10}& \frac{2}{10}& \frac{3}{10}& \frac{4}{10}\end{array}$

Then the variance of X is

The solution set of ${\log }_{1/3}(2^{x+2}-4^x)\ge -2$, is

The principal value of ${\text{cosec}}^{-1}(-2)$ is



[1 mark]

If $b_1,b_2,b_3,\dots$ forms a G.P. and $b_1=1,$ then the common ratio of the G.P. when $4b_2+5b_3$ is minimum, is